An asymptotic on the logarithms of the relative class numbers of imaginary abelian number fields of prime conductor and large degree

TL;DR

This paper derives an asymptotic formula for the logarithms of relative class numbers of certain imaginary abelian number fields with prime conductors and large degrees, and investigates the limits of this asymptotic behavior.

Contribution

It provides a new asymptotic analysis for class numbers of imaginary abelian fields with prime conductors and large degrees, extending previous results.

Findings

Asymptotic formula established for cases where φ(d)=o(log p)

Asymptotic does not hold when φ(d)=O(log p)

Results clarify the limits of asymptotic behavior for class numbers

Abstract

An asymptotic on the logarithms of the relative class numbers of the cyclotomic number fields of prime conductors $p$ is known. Here we give an asymptotic on the logarithms of the relative class numbers of the imaginary abelian number fields of prime conductors $p$ and large degrees $m =(p-1)/d$ with $\phi(d)=o(\log p)$. We also show that this asymptotic does not hold true anymore under the only slightly weaker restriction $\phi(d)=O(\log p)$.